metals

Review

Control and Design of the Steel Continuous CastingProcess Based on Advanced Numerical Models

Katarzyna Miłkowska-Piszczek * and Jan Falkus

Department of Ferrous Metallurgy, Faculty of Metals Engineering and Industrial Computer Science,AGH University of Science and Technology, 30-059 Krakow, Poland; jfalkus@agh.edu.pl* Correspondence: kamilko@agh.edu.pl; Tel.: +48-12-617-2558

Received: 12 July 2018; Accepted: 27 July 2018; Published: 30 July 2018�����������������

Abstract: The process of continuous casting of steel is a complex technological task, including issuesrelated to heat transfer, the steel solidification process, liquid metal flow and phase transitionsin the solid state. This involves considerable difficulty in creating the optimal process controlsystem, which would include the influence of all the physico-chemical phenomena which mayoccur. In parallel, there is an intensive development of new mathematical models and an increase incomputer performance, therefore complex numerical simulations requiring substantial computingtime can be conducted. This paper presents a review of currently applied numerical methods allowingthe phenomena accompanying the process of continuous casting of steel to be accurately represented.Special attention was paid to the selection of appropriate methods to solve the technological problemselected. The possibilities of applying selected numerical models were analysed in order to modifyand improve the existing process or to design a new one linked to the implementation of new steelgrades in the current production. The description of the method of defining the boundary conditions,initial conditions and material parameters as vital components ensuring that numerical calculationsbased upon them in the finite element method, which is that most frequently applied, are correct is animportant element of the paper. The possibility of reliably defining the values of boundary parameterson the basis of information on the intensity of cooling in individual zones of the continuous castingmachine was analysed.

Keywords: continuous casting process; numerical modelling; finite element method; process control

1. Introduction

Modelling the process of steel continuous casting is a very complex task, and can be accomplishedusing various types of mathematical models. The assumed computing objective and the requiredaccuracy should be key in selecting the model. In many cases, the desired information is knowledgeof the metallurgical length of the strand and the dynamics of changes in the shell thickness. This is thecase when determining a place for carrying out the so-called soft reduction operation. As may happenwhen the strand casting speed needs to be changed, a procedure allowing a new cooling intensityto be determined is required. A problem like this does not require answering a series of questionsrelated to stress occurring in the strand, the structure formed, or potential cast strand defects. Thus,it is understandable that the model is naturally simplified to a form, which still provides a credibleanswer to the questions that are primarily asked [1–4].

At the early stage of problem solving, the correct selection of the model type and relatedpossibilities for its adaptation to the class of the solved problem is a difficult challenge. Theoretically,a more complex model (which is to say more “intelligent”) can easily answer questions aboutthe primary technological parameters of the casting process. Yet in practice we encounter a numberof limitations. Assuming hypothetically that a complex model has been verified as correct, in the best

Metals 2018, 8, 591; doi:10.3390/met8080591 www.mdpi.com/journal/metals

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case we face an unnecessary extension of the computing time. It results from the fact that the modelcalculates much more parameters than is needed to solve the defined problem.

The second danger caused by non-synchronising the complexity of the set problem with the“intelligence” of the used tool is the issue of the verification of model parameters and their correlationwith the process data. The more theoretically elaborate the model, the more parameters and the higherrisk of occurrence of unmeasurable parameters. The last comment concerns the problem of the strategyof acquiring knowledge of the value of the required model parameters. Many years of experiencein modelling the continuous steel casting process show that the best choice is an experimentalmeasurement of all measurable model parameters. It can be illustrated by parameters in the form of thespecific heat of the steel cast as a function of temperature, heat conductivity for the steel, viscosity,etc. This paper will present considerations on the method of choosing priorities in planning the basicmeasurements of the properties of the material cast.

2. The Process of Steel Continuous Casting as an Object of Numerical Modelling

The process of steel continuous casting—due to its complexity—is accompanied by many physicalphenomena. The steel solidification process within the mould and after leaving the mould—in thesecondary cooling zone—should fall within the most important of those effects. In the primary coolingzone, the following partial processes should be distinguished [2–7]:

• Turbulent flow of liquid steel through a complex geometry area—a submerged entry nozzle (SEN)or a shroud—caused by convection

• Heat transfer within the liquid steel area• Heat transfer in the mould between the forming shell and the mould wall• Heat flow through the layer of solid and liquid slag• Formation of thermal stress• Shrinkage of the solidifying shell, related to transitions occurring during the steel

solidification process• Thermal effect accompanying the solidification phenomenon• Mechanical impact of the mould walls on the solidifying strand• The process of an air gap formation between the mould wall and the solidifying strand• The formation of crystals within the solidification zone accompanied by element

segregation effects• Formation of surface defects

In the secondary cooling zone, the following processes should be distinguished:

• Heat transfer within the liquid core area (conduction and convection)• Heat conduction in the solidified shell layer• Thermal effect accompanying the solidification phenomenon• Multi-stage heat transfer resulting from the strand cooling by the nozzle system, related to the

number of spray zones and the applied cooling type• Shrinkage of the solidifying strand, related to transitions occurring during the steel

solidification process• Formation of individual solidification zones (zone of dendritic crystals and zone of

equiaxed crystals)• Formation of stress related to the contact of rolls with the strand, and the possibility of bulging

between the continuous casting machine rolls

At present it is not possible to simultaneously capture all of the above effects occurring during thewhole process of steel continuous casting, and to present them in the form of a single comprehensivenumerical model. The natural division applied in the continuous casting process modelling is related

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to an attempt at identifying the ensuing problem during actual steel casting, or focusing on a selectedsection of the process in order to improve the existing technology. The use of results of numericalmodelling of the continuous casting process in the industrial practice is most often related to theidentification of defects forming during the steel casting process. A good example of the effectiveapplication of numerical models is the elimination of a problem concerning strand corner cracksimmediately after leaving the mould for the strand size 240 × 240 mm, at the steelmaking shopof Ascometal, France [8]. Numerical simulations with the commercial software package THERCAST®

have established the cause of the cracks. The analysis of the calculated stress distribution, along withthe applied Yamanaka fracture criterion, has resulted in developing a specific recommendation to addan additional pair of rolls or to replace the mould with a higher one.

The starting point of the numerical modelling of the continuous casting process—includingeffects occurring from the time of filling the tundish until leaving the secondary cooling chamberby the solidified strand—is the calculation of the correct temperature distribution. It is the basis forall numerical calculations and the starting point for conducting further considerations concerningthe creation of patterns of liquid metal flow in the mould, assessment of the design and submersionlevel of the SEN, computing thermal stress occurring within the mould, predicting cast strand defects,simulations of the cast strand microstructure, assessment of the influence of design changes in themain structural components of the continuous casting machine, or changes in the technologicalparameters [1–11].

The vast majority of studies on numerical modelling of the continuous casting process is basedupon the Finite Element Method (FEM) and Finite Volume Method (FVM). These methods are basedon replacing a continuous description of the temperature fields analysed, stress state, etc., with anapproximate discrete description. To obtain results from a numerical simulation, systems of algebraicequations are used. Their solutions serve for determining a discrete (discontinuous) approximationof the continuous solution of differential equations describing effects occurring in metals and alloysduring solidification (heat flow, fluid movement, phase transitions, processes of structure formation,and others) [1,2,5,9–11]. The commercial packages ANSYS Fluent® and PHOENICS® [12–14] are mostoften used for numerical modelling of effects related to the liquid steel flow in the tundish. They arebased upon numerical methods and algorithms applied in Computational Fluid Dynamics (CFD) [15].

Research conducted by many authors [12–14,16–21] shows a very good compliance betweensimulations of the flow of liquid steel through the tundish, using turbulent flow models k-ε,and tests performed on water models and tests in industrial conditions. Most numerical calculationsare conducted on the basis of the Reynolds-averaged Navier-Stokes equations (RANS) method,which allows the average rate of turbulent flow in the tundish to be determined. However,the application of the Large Eddy Simulation (LES) method, along with the Smagorinsky-Lillyturbulent model, allows us to determine the instantaneous velocity field of the steel inside thetundish [12,16,18,22]. Not only are the selection of the method of computing the rate of liquid steelflow through the tundish and the selection of the main model parameters key to the values of liquidsteel velocity vectors in individual nodes of the finite element mesh, but they are also crucial to thetemperature distribution within the whole tundish volume.

Computing made within the primary cooling zone is related mostly to the analysis of geometry,position and submersion depth of the SEN. The influence of the liquid steel turbulent flow from thetundish on the temperature distribution within the mould, and thus on the formation of the shell,is investigated. One should stress that obtaining a proper shell thickness after leaving the mould iskey to the safety of the steel casting process. Most failures encountered during the actual continuouscasting process are related to breakouts within the primary cooling zone. Ensuring a sufficient shellthickness is also related to the appropriate selection of the value of the mould wall taper, which is tocompensate for the effect of steel shrinkage and to limit the occurrence of a gaseous gap. Many authorsattempt to model effects occurring within the primary cooling zone using both commercial packagessuch as ANSYS Fluent®, ProCAST® and THERCAST® [3–6,12–14,16–20], and their own original codes

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based on the FEM [1,2,11,23–26]. Numerical modelling of the primary cooling zone also includes anissue related to computing the thermal stress, which combined with a correct fracture criterion may beused for determining the correct mould wall taper in correlation with the casting speed.

At present, more and more numerical simulations of the continuous casting process are focusedon taking into account the influence of additional casting support equipment e.g., electromagneticstirrers (EMS) or the soft reduction system. Most studies—concerning a description of the Lorentzforce impact on the liquid steel movement—are related to a numerical simulation of the influence ofan electromagnetic stirrer or brake on the flow of the liquid steel in the mould [27–30]. The solution tosuch a problem is possible by computing the temperature distribution, along with the flow of liquidsteel, and additionally taking into account the convection in the liquid steel generated by the impactof the electromagnetic force. One should emphasise that numerical models allowing the Maxwellequations to be solved are extremely complex, and often they require lengthy computing time. In thisconnection, computing the full temperature distribution including the effect of electromagnetic stirrerswithin the secondary cooling zone e.g., for a system of two S-EMS (strand electromagnetic stirrers) andF-EMS (final electromagnetic stirrers) on a 20 m long strand, which would allow the actual technologyto be fully reproduced, remains problematic.

A strand cast obtained in the process of steel continuous casting is characterised by a diversecrystalline structure consisting of the chill zone, columnar zone and equiaxed zone. Contemporarynumerical solutions use the Cellular Automata (CA) method to model processes of formation of thecast strand structure, taking nucleation and grain growth into account [31]. An attempt to modelprocesses of formation of the cast strand structure, including nucleation and grain growth, was madein the studies [32,33]. A micro-macro type numerical model was used in these studies. In the micro-macro modelling the issue of heat flow is investigated on the scale of the whole strand (macro).The mathematical model of nucleation and grain growth on the micro scale allows us to describethe process of structure formation and to determine the crystallisation heat emission rate. Modelling theinternal structure of dendritic grains in a two-dimensional space with the cellular automata methodis presented in [34]. The paper [34,35] presents the capabilities of a model like this to describe themechanism of forming a narrow area with an increased sulphur content in the strand immediatelyunder the chill zone. However, as a cellular automaton with a mesh of 1 µm must be applied, it is notsuitable to model crystallisation in large spaces. The Cellular Automata Finite Element (CAFE) model,which is now available as a ProCAST® software module, was used in the papers [32–35] to describethe process of nucleation and grain growth of the primary phase in the cast strand. In this modelling,mathematical methods of cellular automata are used. The cellular automata method enables boththe stochastic grain nucleation on the strand surface and within its volume to be simulated, and thegrowth of individual primary phase grains to be analysed. The modelling results in the historyof changes of the interface position, and the determination of the grain boundaries after completionof the strand solidification. The CA method computations are coupled with the simulation of thestrand temperature field.

3. Development of the Numerical Model for the Continuous Casting of Steel Process

Describing the heat transfer model in the continuous steel casting process is an extremelycomplex task, as it involves all three mechanisms of heat transfer: conduction, radiation andconvection [1–7,36–38]. The following processes influence heat transfer in the continuous steel castingprocess: conduction and convection in the liquid steel, conduction in the solidified shell, heat transportbetween the outer layer of the solidified shell and the mould wall surface through the air gap formingwithin the mould, heat conduction within the mould, heat transfer within the mould between thechannel walls and the cooling water, heat transfer within the secondary cooling zone by convectionand radiation, heat transfer between the solidifying strand and the rolls by conduction. Additionally,thermal effects related to the accompanying phase transitions have a significant impact on the heattransfer model.

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Discretisation of the domain modelled is the first step in creating an FEM model. It is breakingdown a given domain into finite elements, which are elements of a FEM mesh. The elements sharenodes and additionally they contain information on their mutual neighbourhood. Discretisation isthe key issue, as the correctness, accuracy and numerical computing time depends on the quality of thefinite element mesh. Increasing mesh density extends the computing time and leads to an increase indemand for the size of the required operating memory used during computing, the number of carriersrequired for archiving the results, and the result access time. However, by increasing the mesh densitya higher accuracy of the obtained simulation results is achieved.

The basic element of an irregular computing mesh, which should be applied when modellingthe majority of typical casting processes, is a linear four-node element—tetrahedron. A tetrahedronmesh is generated automatically on the basis of the surface mesh, with a triangle as its main element.The possibility and often the necessity of optimisation of the created finite element mesh is an importantaspect. Most often it involves refinement of the mesh in the area of the boundary condition occurrence.For continuous casting of steel, those will primarily be the areas at the mould walls and on the strandsurface within the secondary cooling zone. Generally, as a rule, the mesh should be refined (using ashorter side length) in the areas where during the problem solving the highest temperature gradientsare expected.

3.1. Strategy of the Boundary Conditions Calculation

The temperature field can be determined by solving the generalised Fourier equation, which alsodescribes the heat transfer. In the general form this equation is written as follows [3–6]:

∂(ρcpT)∂t

=∂

∂x(λ

∂T∂x

) +∂

∂y(λ

∂T∂y

) +∂

∂z(λ

∂T∂z

) + Q (1)

where ρ is density, kg·m−1; cp is specific heat, kJ·kg−1·K−1; λ is thermal conductivity, W·m−1·K−1;t is time, s; T is temperature, K; Q is the heat source term, W·m−3; x, y, z are 3D coordinateaxes. The solution to the thermal problem is the T vector, representing the temperature valuesin the individual nodes of the finite element mesh. The solution to the generalised diffusionequation—the Fourier equation—should meet the boundary conditions declared on the cast strandsurface. The boundary conditions may be described with a general equation in the form [3–6]:

− λ∂Twl∂n

= α(Twl − Ta) (2)

where Ta—temperature of the surrounding medium, K; Twl—temperature of the cast strand surface, K;α—heat transfer coefficient (HTC), W·m−2·K−1.

Obtaining a unique solution to the heat transfer equation involves declaring the boundaryconditions of the heat transfer. In the source literature the boundary conditions are divided into threebasic groups:

• The first-type boundary condition, when the surface temperature at the edge of the domainis specified.

• The second-type boundary condition, when the heat flux density transferred to the environmentat the edge of the domain is specified.

• The third-type boundary condition, when the method of heat transfer at the edge of thedomain expressed numerically by the heat transfer coefficient, and the ambient temperatureare determined.

The biggest difficulty as regards computing reliable boundary conditions is related to the selectionof a model describing the heat transfer in the primary and secondary cooling zones, and to the collectionof a database of actual process parameters. The contact of the face of the solidifying cast strand withthe inner side of the mould and the heat transfer accompanying this process can be implemented by

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determining the value of the heat transfer coefficient as a function of the strand surface temperature.The faces of the 3D model with the applied finite element mesh, for which the heat transfer coefficientshould be computed, can be divided into four groups:

1. The mould outer side2. The contact of the solidifying strand face with the inner side of the mould3. The surface of the liquid steel meniscus4. The secondary cooling zone (divided into a set number of spray zones)

The authors of the papers [3–6] provide average values of the heat transfer coefficient from 18,000to 24,000 W·m−2·K−1, which corresponds to the value of heat received by the water flowing withinthe mould channels—the outer side of the mould. The above value was calculated on the basis ofprocess data related to the cooling water temperature at the inlet and outlet of the mould, and therate of cooling water flow for each of the mould walls (slab casting machine, size 220 × 1100 mm).The share of the water-cooled mould surface is an essential parameter taken into account in computing.

The boundary condition related to the contact of the solidifying strand face with the mould walls isamongst the most difficult to determine. Many models have been presented in the literature. Basically,they can be classified into the following groups: heat transfer models based upon the average valueof heat transfer coefficient, and models presenting a functional relationship between the strand surfacetemperature and the heat transfer coefficient. In addition, heat transfer models may assume that an airgap forms during casting. The air gap preventing contact of the solidifying strand with the mould walldisturbs the heat transfer process. In the paper [6], the influence of a change in the coefficient of theheat transfer between the strand and the mould was examined in detail by calculating the temperaturedistribution in the primary and secondary cooling zones. The authors compared 11 literature modelsand developed a new one, empirical model (No. 12) of heat transfer between the solidifying strandand the mould wall, assuming the occurrence of an air gap. The authors analysed models based uponthe selected average values of heat transfer coefficient—group 1, next models taking into accounta step change in the value of the heat transfer coefficient—group 2, and two cases, where the heattransfer coefficient changed linearly—group 3. The biggest group—group 4—comprised complexmodels allowing the temperature distribution to be calculated for the heat transfer coefficient as afunction of the strand surface temperature. Data concerning the average and maximum values of theheat transfer coefficient is presented in Table 1.

Table 1. The heat transfer coefficient in the primary cooling zone—a comparison of the models (adoptedfrom [6], with permission from Polish Academy of Science, 2018).

Number of the HTC Model Type of the HTC Model Average or Maximum Value ofthe HTC [W·m−2·K−1]

1 The average value of the HTC HTC = 12002 The average value of the HTC HTC = 13003 The average value of the HTC HTC = 1500

4 Two values of the HTC HTC = 1163 for z ≤ 0.6 mHTC = 1395.6 for z > 0.6 m

5 A linear values of the HTC HTC = 800–20006 A linear values of the HTC HTC = 600–15007 The values of the HTC as a function HTCmax = 13008 The values of the HTC as a function HTCmax = 25009 The values of the HTC as a function HTCmax = 2000

10 The values of the HTC as a function HTCmax = 130011 The values of the HTC as a function HTCmax = 309712 The values of the HTC as a function HTCmax = 1600

z—height of the mould (m).

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The tests showed that the maximum difference for the temperature measured at the surface of thestrand leaving the mould between the models (models No. 3 and No. 10) investigated was as high as327 ◦C. The foregoing value shows a discrepancy that may occur when the various type of the heattransfer in the primary cooling zone are applied. In order to calculate the maximum difference for thetemperature one measured point, at the surface of the strand was selected, for all implemented heattransfer models. The authors of the above-mentioned paper also compared the thickness of the shellleaving the mould and obtained about 1 cm difference between the extreme variants (models No. 3and No. 10). Table 2 presents the results of simulations conducted for the HTC values presented inTable 1.

Table 2. Comparison of the shell thickness and temperature for those selected models of HTC (adoptedfrom [6], with permission from Polish Academy of Science, 2018).

Number of the HTC Model Thickness of the Shell afterLeaving the Mould, cm Temperature, ◦C

1 2 8362 2.3 8123 2.75 7724 2.27 8335 2.38 9006 1.94 9567 1.98 9768 1.92 10509 1.86 1090

10 1.82 109911 2.51 87412 2.52 938

The obtained results clearly show that the HTC is a very important parameter of the continuouscasting model and lack of information about the scale of its impact leads to significant deviationsof model calculations from the actual strand temperature distribution. The results of the conductednumerical simulations have also shown the importance of proper selection and verification of theadopted concept of description of the heat transfer within the continuous caster mould.

If the solidifying strand surface temperature is not known, a simplified formula may be appliedto calculate the heat transfer coefficient for individual spray zones within the secondary coolingzone [3–6]:

αspray = 10v + (107 + 0.688v)w (3)

where v is water drop velocity, m·s−1; w is water flux density, dm3·m−2·s−1. For each of the sprayzones the average value of heat transfer coefficient can be computed on the basis of water flux density.It does not take into account places of contact of rolls with the strand. The main technologicalparameter—necessary to compute the heat transfer coefficient—is the cooling intensity, most oftengiven in dm3·min−1, which divided by the area of an individual spray zone allows the water fluxdensity to be computed for the selected casting speed. In the literature [1,2,7,11,39–43], we can findmany models, which allow the heat transfer coefficient to be computed in a selected spray zone as afunction of the solidifying strand surface temperature.

One needs to emphasise that for the primary and secondary cooling zones there is no singleuniversal heat transfer model, which would allow the correct value of heat transfer coefficient to becomputed. Each time when a selected heat transfer model is implemented, it needs to be adjusted tothe actual conditions of the process, starting with the mould type, its height and the place of the air gapformation. The database containing technological data on cooling in the continuous casting process iskey to computing the values of the declared boundary conditions. It must contain data concerningwater flow rates within individual spray zones and flow rates through the mould walls, cooling water

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temperature both for the primary and the secondary zone. The difference in temperature betweenthe incoming and outgoing water from the mould cooling channels is very important information forcomputing the thermal balance. For the secondary cooling zone, it is necessary to analyse the flowrates of the cooling water in the individual spray zones in correlation to the casting speed. In order toformulate a reliable model of heat transfer within the secondary cooling zone, also information on thecooling nozzles, their number, spray angle and diameters in the individual spray zones will be needed.

3.2. Material-Related Parameters

Chemical composition is the basic feature that determines the classification of steels. It defines boththe casting process conditions and the other thermophysical and rheological properties of the materialtested. The liquidus temperature is calculated on the basis of the steel chemical composition with avery good accuracy. Well known formulas which are applied in industrial practice, e.g., the Roeser andWensel formula, can be used for this purpose. Also experimental methods (e.g., Differential ScanningCalorimetry—Differential Thermal Analysis analysis) or thermodynamic databases can be applied. Thelatter can be used to determine other material-related parameters. The Fourier equation, Formula (3),must be complemented with the following thermophysical parameters: specific heat or enthalpy,thermal conductivity, density. To calculate the temperature distribution in the continuous castingprocess it is also necessary to know the solidus temperature, while the value of the viscosity of theselected steel grade is necessary for modelling the flows. The movement of liquid steel is described byNavier-Stokes equations resulting from the momentum balance, supplemented with mass conservationequations [1,2,11,44]. They contain both the dynamic viscosity coefficient and the kinematic viscositycoefficient [45–47]. Modelling the stress distribution during casting involves the need to declare therheological parameters i.e., the Young modulus, the Poisson’s ratio, yield stress, ultimate yield stress,plastic strain and thermal expansion.

The correctness of the numerical calculation results strictly depends on the correctness andaccuracy of material-related data implemented in the model. The specific heat, along with the enthalpyaccompanying transitions occurring within the solidifying cast strand, are amongst the most importantparameters. The relationship between enthalpy and specific heat, at a set temperature for metal alloysis defined as follows [3–6,48]:

H(t) =t∫

t0

cp(t)dt + (1 − fS)L tsol ≤ t ≤ tliq (4)

where cp is specific heat, kJ·kg−1·K−1; L—latent heat, kJ; fS—solid fraction (0–1). Formula (4) presentsalso the method for declaring the latent heat between the liquidus and solidus temperatures.

It should be emphasised that specific heat values can be implemented in the generalised Fourierequation (Equation (1)), as well as the values of enthalpy calculated on the basis of specific heat values.The enthalpy approach allows the heat accompanying transitions occurring in the solidifying steel tobe presented accurately. Paper [44] presents a numerical model of the primary cooling zone with asubmerged entry nozzle. A series of numerical calculations were performed to show the differencein the temperature distribution within the mould zone for the declared values of specific heat andenthalpy. It was shown that the method of implementation of the values of specific heat and enthalpyin the numerical model of the continuous casting process is key to the correct calculation of thesolidifying strand temperature distribution and the prediction of shell thickness. The differencesin the temperature distribution in the primary cooling zone for the selected values of specific heatand enthalpy (including values computed with thermodynamic databases) were as high as 200 ◦C.The presented results showed the influence of material-related values implemented in a numericalmodel on the final results.

The authors of papers [10,24,25,49–52] apply average values of specific heat in modelling thecontinuous casting process. It seems to be a good trade-off when no experimental data is available.

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To compute the values of enthalpy as a function of temperature, thermodynamic databases can be used,e.g., being an integral part of the Thermo-Calc® or LCC Computherm® software databases suppliedtogether with ProCAST® software. It is a frequently applied approach in designing numerical models.It is recommended if the results of experimental tests are obtained at a temperature lower than thesolidus temperature e.g., for tests of thermal conductivity. Material-related parameters computedwith thermodynamic databases can complement experiments conducted to determine the functionalrelationship of temperature and the values of the thermophysical parameter. It should be emphasisedthat the reliability of material-related parameters is another extremely valid factor which influencesthe correctness of numerical computing results.

3.3. Inverse Modeling

In Sections 3.1 and 3.2 a set of necessary boundary conditions is presented, along with thethermophysical parameters that are implemented in the numerical model of the continuous castingprocess. The determination of their values, most often as a function of temperature, is a time consumingand expensive task. The simplest method for defining the set of model input data is to use the literaturedata. Unfortunately, the available data concerns a limited number of materials for steels they arepractically limited to those steels that were developed in the 1970s at the latest. Additionally, it isvery difficult to find data concerning material tests conducted at very high temperatures—close tothe solidus temperature of the steel grade tested. The problem is additionally complicated by a broadrange of steel grades that are manufactured nowadays and the high cost of some material tests.Experimental methods allowing specific heat, viscosity, or thermal conductivity to be determinedrequire developing a measurement methodology and carrying out multiple measurements—includingcalibration measurements.

A similar problem appears when we implement data related to the boundary conditions inthe numerical model—with heat transfer coefficients within the primary and secondary coolingzones. Very often, access to the full characteristics of the continuous casting machine or industrialdata recording casting parameters for a selected sequence is hampered. Note that part of the controldata—primarily changes made by the machine operator during casting (adjustment of cooling intensity,emergency mode casting, etc.)—is often not recorded together with the control data. A difficulty indetermining the boundary conditions is also related to developing and adjusting the selected heattransfer model to a specific technological solution.

When the boundary conditions cannot be determined in a classic way—directly from the processdata—it is possible to use inverse analysis to find the searched heat transfer coefficient. In the Inversemethod, contrary to the direct solution, where the equation parameters are known, the input dataare the results of the conducted tests. The main objective of the Inverse method is to determinethe missing model parameters or the range of their changes. Inverse methods use temperaturemeasurements at a single or a few points of the volume of a sensor either heated or cooled at a knowninitial temperature distribution. The boundary condition is searched on the basis of the changes oftemperature. After substituting the boundary condition into the analytical or numerical solutionof the heat conduction equation, the temperature field is achieved that is the best reflection of thechanges measured during the experiment. It is often defined by the heat transfer coefficient betweenthe interface and the environment. When the ambient temperature is known, it is easy to compute theequivalent condition expressed by the heat flux or its density. The Inverse method can be applied todetermine the heat transfer coefficient between the mould wall and the solidifying strand thanks to theknowledge of the readings of temperature values—most often at places of thermocouple installation inthe mould walls. At present, elaborate detection systems for shell adhesion in the primary coolingzone are applied in continuous casting machines. They are based upon continuous monitoring ofthe mould wall temperature at reference points. In modern moulds, systems allowing temperatureto be monitored along the whole wall height are installed—both for wide and narrow walls—alongwith computing the density of heat flux received from the solidified steel. The temperature data

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obtained this way provides a very good foundation to determine the average value of the heat transfercoefficient (or its values as a function of temperature) within the zone of contact of the strand with themould wall.

At present, the authors of papers [53–55] in their computing based on the Inverse method, use boththeir original source codes and commercial computing packages. The presented papers show thatusing the inverse analysis in the numerical modelling of the continuous casting process is very widelyapplied, due to the possibility of precise determination of boundary conditions and the selectedmaterial-related parameters.

4. Sensitivity Analysis

The subject of the sensitivity analysis is to investigate the impact of input parameters on thevariability of output parameters of a model. The definition formulated this way allows us to introduceclassification, which enables the sensitivity analysis to be adjusted to the needs of the continuous castingmodelling. Formally, the analysis of possibilities should answer, for instance, the following questions:

• What factors cause the biggest changes in the metallurgical length of the strand and the shellthickness under the mould?

• Are there any factors whose influence on the above mentioned parameters is negligible?• Are there any interactions, which enhance or suppress the variability caused by individual factors?

Formally, various kinds of sensitivity analysis can be distinguished depending on the methodused to obtain an answer to the question asked above. Unlike the continuous casting process models,we define the key division of these methods as local or global sensitivity analysis. Local sensitivityanalysis takes into account output variability related to the changes of input parameters around thenominal value determined. On the other hand, global sensitivity analysis takes into account changes inthe whole space of the variability of input parameters. With respect to the continuous casting processmodelling problem concerned, the local sensitivity analysis is much more useful. It arises from the factthat for the verification of the process model, industrial data of an actual continuous casting machineis most often used, for which nominal values of the model input parameters are approximately known.An attempt to use the global method involves the necessity to define the scope of acceptable variabilityof all input parameters. This may cause errors that are difficult to estimate.

The problem of sensitivity analysis of the model applied to simulate the continuous steel castingprocess needs to be discussed in detail. The classic approach to the problem, known for example fromthe economy field, is of little use to the continuous steel casting process. To clearly define the problem,one should distinctly emphasise that when talking about the sensitivity of a continuous casting processmodel, it is recommended to take into account the sensitivity analysis both with regard to the controlparameters and the process state parameters, as well as model parameters determined at the stage ofits verification. Improper care in both mentioned cases very often leads to the occurrence of criticalerrors in the model, disqualifying the possibility of its application. The research findings discussed inSection 3.1, presented in Table 1, are a good illustration of this problem.

The analysis of various mathematical models of the continuous steel casting process allows us topropose the following classification of their parameters.

I. Technological process parameters: strand casting speed and cooling intensity in individualzones of the continuous casting machine.

II. Parameters characteristic of the properties of the material cast:

• chemical composition of the steel cast, (%)• initial temperature of the steel cast (in the tundish), (K)• liquidus temperature of the steel cast, (K)• solidus temperature of the steel cast, (K)

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• viscosity of the steel cast as a function of temperature, (mP)• specific heat of the steel cast as a function of temperature, (J·kg−1·K−1)• melting heat of the steel, (J·kg−1)• steel density as a function of temperature (kg·m−3)• thermal conductivity, (W·m−1·K−1)

Before starting a detailed analysis of the mentioned parameters regarding their impact on the finalresult of computing, note that determining the values of some of them is very precise even in industrialconditions. It means that conducting sensitivity analysis for those parameters is not appropriate, as theproblem does not concern them. In the first group of technological parameters, the strand casting speedis an example of a parameter that is determined very accurately. With the current state of measurementtechnology in this area, one may assume that the accuracy of determination of the strand withdrawalspeed is absolutely sufficient for model calculations to be accurate.

The second of the mentioned control parameters is cooling intensity, which in each model isrepresented by the boundary conditions. In practice it is one of the most difficult problems relatedto the mathematical modelling of the continuous casting process. There may be various methods ofconverting the intensity of the supplied cooling medium to the heat transfer coefficient HTC. However,in any case, they constitute a potential source of errors. Such a state simply forces the need to assess thesensitivity of the continuous casting process model applied to the value of the heat transfer coefficientHTC [5,6].

In the second group defined as “material-related properties of the steel cast”, the only parameterwhich can be very accurately measured is the temperature of the steel cast in the tundish, which istreated as the initial value for the conducted simulations of the casting process. Note that the valueof this temperature may fluctuate and the sensitivity analysis for this parameter may only be omittedif a reliable system of steel temperature measurement in the tundish is in place.

For other parameters of the second group, classic tests based upon the local sensitivity analysismethod were conducted. The justification of the selection of this particular method has already beenprovided above. The results of the conducted analysis allowed us, as expected, to sort all parametersby the criterion testing their influence on the final result of computing the shell thickness and thestrand metallurgical length.

Summing up the conducted sensitivity analysis of continuous steel casting process models,one should state that three parameters are the most important in terms of sensitivity:

• HTC in the mould• HTC in the secondary cooling zone• Steel specific heat as a function of temperature cp(T)

5. Evaluation of Possibilities for Using Numerical Models in Order to Design or Modify theExisting Technology

The necessary condition for the correct selection of parameters of strand cooling within thecontinuous caster is to have a reliable process model, taking into account the design of the machineand its relevant technical parameters. A verified and tested model allows changes in the existingcasting process to be made and processes for new steel grades to be engineered.

5.1. Development of the Cooling Program for a New Steel Grade

The method of creating a process manual for a specific steel group includes determining thecooling intensity in the individual zones of the machine for a standard casting speed and modificationof the cooling intensity for a changed speed. A set of information about cooling water intensityin machine sections depending on the casting speed is called a cooling schedule. The strategy ofdeveloping a new cooling schedule for a selected steel grade, assuming the existence of a correctly

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verified process model, can be accomplished according to, for instance, the procedure presented below.It comprises six stages:

• The simulation of the casting process for various strand withdrawal speeds, taking into accountthe technical parameters of the continuous casting machine without changing the values of thecooling parameters.

• The assessment of the impact of a speed change on changes of basic parameters of the strand casti.e., the metallurgical length, shell thickness under the mould, and the value of the strand surfacetemperature at the reference points.

• The estimation of the necessary change in values of heat transfer coefficients in the individualcooling zones of the continuous casting machine allowing the assumed metallurgical length andshell thickness under the mould to be achieved for the changed casting speed.

• The conversion of new heat transfer coefficients to water consumption in individual secondarycooling zones.

• The experimental check of the correctness of the calculated temperature of the strand surface atthe selected reference points.

5.2. Method of Selection of Cooling Parameters

In Section 5.1, an algorithm of proceeding during the development of a new cooling schedulewas proposed. It requires, among others, computing the necessary change of the values of the heattransfer coefficients in the individual cooling zones of the continuous casting machine, to guaranteethe assumed metallurgical length. The length of the liquid core—metallurgical length—measuredfrom the level of liquid steel meniscus in the mould to the time of total solidification of the caststrand—is a very important parameter in the continuous casting process. Often maintaining a similarmetallurgical length for various casting speeds is a challenge during the actual steel casting process.In industrial practice, it is aimed at total strand solidification at the place where it leaves thesecondary cooling chamber or exactly at the point of the mechanical “soft reduction” zone. In addition,maintaining comparable metallurgical lengths for various strand casting speeds guarantees that safetyis ensured during the continuous casting process as the strand is fully solidified before shearing.Another important aspect is the strand straightening points within the secondary cooling zone. It isalso important that the liquid fraction is left at their position. In the papers [3,6], the authors presentedthe utilisation of a numerical model of the continuous steel casting process for determining newcooling parameters in the secondary zone by optimisation of the liquid core length. The methodpresented in the paper is based on determining the percentage contribution of the casting speed intothe metallurgical length at constant cooling rates. Next, on the basis of the heat transfer coefficients foreach of the spray zones, the mean heat transfer coefficient is computed for the whole secondary coolingzone. The next stage is computing the new, average heat transfer coefficient, and the transition toaverage values of the coefficient within the spray zones, and thus the flow rates within individual sprayzones. Confirming that the proposed method is right has been proven by computing the temperaturedistribution in the primary and secondary cooling zones. The template of the described solution,for the secondary cooling zone comprising seven spray zones, is presented in Figure 1.

One should stress that the correctness of model calculations was accurately verified, and basedupon the following parameters: thickness of the shell leaving the mould, strand metallurgical length,and comparing the computed temperature values at the measurement points with the values measuredwith optical pyrometers in the secondary cooling zone. Additionally, the authors presented a solution,which could be directly applied in the industrial conditions thanks to determining new cooling waterflow rates within the secondary cooling zone.

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Metals 2018, 8, 591 12 of 16

5.2. Method of Selection of Cooling Parameters

In Section 5.1, an algorithm of proceeding during the development of a new cooling schedule

was proposed. It requires, among others, computing the necessary change of the values of the heat

transfer coefficients in the individual cooling zones of the continuous casting machine, to guarantee

the assumed metallurgical length. The length of the liquid core—metallurgical length—measured

from the level of liquid steel meniscus in the mould to the time of total solidification of the cast

strand—is a very important parameter in the continuous casting process. Often maintaining a similar

metallurgical length for various casting speeds is a challenge during the actual steel casting process.

In industrial practice, it is aimed at total strand solidification at the place where it leaves the

secondary cooling chamber or exactly at the point of the mechanical “soft reduction” zone. In

addition, maintaining comparable metallurgical lengths for various strand casting speeds guarantees

that safety is ensured during the continuous casting process as the strand is fully solidified before

shearing. Another important aspect is the strand straightening points within the secondary cooling

zone. It is also important that the liquid fraction is left at their position. In the papers [3,6], the authors

presented the utilisation of a numerical model of the continuous steel casting process for determining

new cooling parameters in the secondary zone by optimisation of the liquid core length. The method

presented in the paper is based on determining the percentage contribution of the casting speed into

the metallurgical length at constant cooling rates. Next, on the basis of the heat transfer coefficients

for each of the spray zones, the mean heat transfer coefficient is computed for the whole secondary

cooling zone. The next stage is computing the new, average heat transfer coefficient, and the

transition to average values of the coefficient within the spray zones, and thus the flow rates within

individual spray zones. Confirming that the proposed method is right has been proven by computing

the temperature distribution in the primary and secondary cooling zones. The template of the

described solution, for the secondary cooling zone comprising seven spray zones, is presented in

Figure 1.

Figure 1. Models of heat transfer coefficient versus temperature on the strand surface (adopted from

[3], with permission from Polish Academy of Science, 2018).

One should stress that the correctness of model calculations was accurately verified, and based

upon the following parameters: thickness of the shell leaving the mould, strand metallurgical length,

and comparing the computed temperature values at the measurement points with the values

measured with optical pyrometers in the secondary cooling zone. Additionally, the authors presented

a solution, which could be directly applied in the industrial conditions thanks to determining new

cooling water flow rates within the secondary cooling zone.

6. Conclusions

The control of the steel continuous casting process is one of the most difficult tasks in the steel

production line. The conditions assumed in the casting process, influence both the safe process course

and the required product properties. Evaluating the problem from a historical perspective, we notice

that the advantages resulting from the application of the continuous casting process have been

HTC 1 HTC 2 HTC 3 HTC 4 HTC 5 HTC 6 HTC 7

HTC' 1 HTC' 2 HTC' 3 HTC' 4 HTC' 5 HTC' 6 HTC' 7

SECONDARY COOLING ZONE

HTC average

HTC' average

SECONDARY COOLING ZONE

Figure 1. Models of heat transfer coefficient versus temperature on the strand surface (adopted from [3],with permission from Polish Academy of Science, 2018).

6. Conclusions

The control of the steel continuous casting process is one of the most difficult tasks in the steelproduction line. The conditions assumed in the casting process, influence both the safe processcourse and the required product properties. Evaluating the problem from a historical perspective,we notice that the advantages resulting from the application of the continuous casting process havebeen obvious long before. Technical problems and imperfections of available materials have preventedthe implementation of the expected solution for a long time. From the time when H. Bessemer wasgranted the patent for continuous steel casting in 1846 to the beginning of the 1970s about 130 yearspassed. This was the time needed to overcome the technical challenges and to roll out the process onan industrial scale.

Controlling a modern steel continuous casting process requires full control of the strandsolidification process. The measurement system of a continuous casting machine provides a lotof process information. However, key information, such as changes in the shell thickness at individualpoints of the machine and the metallurgical length (the length of the liquid core), is missing. It arisesfrom the described reasons that the mathematical model of the process is extremely important inthe control system of the continuous casting process. Its accuracy allows it to be used in makingtechnological decisions.

Observing the development of scientific research on the described problem, one can show twocompatibility groups which characterise current art and the level of practical solutions. The first groupconcerns the selection of the type of model describing the continuous casting process. Except forthe differences resulting from the adopted specific solutions, one needs to state that models basedupon the finite element method prevail. The possibilities of models of this class seem to be sufficientin solving any existing or future challenges. The other group of problems seem to be not fullysolved yet. Generally, it is about a reliable transformation of essential process parameters to theboundary conditions of a mathematical model. Indeed, the existing solutions with this regardalready provide satisfactory results, but there is still plenty of room for the continuation of research,especially in the experimental domain. An additional motivation to carry this research out is thesensitivity of continuous casting process models to the values of HTC, established beyond any doubts.Other parameters, whose significance was shown by the sensitivity analysis, should be determinedexperimentally. Moreover, the use of inverse analysis is very helpful. As computer capacity increases,it becomes more and more important.

Nowadays, the development of numerical models of the continuous casting process definitelyimproves the ability to predict cast strand defects and the possibilities for determining the influenceof additional devices, such as electromagnetic stirrers or the location of the “soft reduction” zone.The right development trend includes tests conducted on modelling the natural effect of elementsegregation during continuous casting. Any studies using numerical models, which contributedto the development of technological guidelines resulting in an increase in steel product quality,

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the elimination of steel semi-product defects, and reducing the risk of carrying out the so-called hottests, properly justify ongoing research on the numerical modelling of the continuous casting process.

Author Contributions: K.M.P. and J.F. prepared materials and wrote the paper.

Funding: This research received no external funding.

Acknowledgments: Research financed through statutory funds of AGH University of Science and Technology inKrakow, No. 11.11.110.293.

Conflicts of Interest: The authors declare no conflict of interest.

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